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Engineering Mathematics

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Published in: Mathematics
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Cayley-Hamilton Theorem and its applications

Devi / Chennai

6 years of teaching experience

Qualification: B.Tech/B.E. ( Saveetha Engineering College, Chennai - 2017)

Teaches: Basic Computer, MS Office, School Level Computer, Biology, Physics, English, EVS, Mathematics, Computer, Instrumentation, C / C++

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  1. Cayley Hamilton Theorem: Every Square matrix satisfies its own characteristic equation. Uses: To calculate (i) The positive integral powers of A and (ii) The inverse of a square matrix A. 8A2 + 3 A 1. Verify that A Solution: 2 1 1 1 2 1 2 4 1 satisfies its own characteristic equation and hence find A 2 For the given matrix DI — 6, D —8, D 3 The characteristic equation is given by 13 — 612 By Cayley Hamilton theorem, A 3 — 6A2 + 8A From the given matrix +81 3 31 -o 7 5 5 6A2 -k 814 6 6 5 31 o o o 9 7 29 22 22 o o o 28 23 22 29 22 22 38 28 29 28 23 22 6 7 5 5 38 28 29 6 6 5 9 6 7 6A3 +8 2 1 1 1 2 1 2 1 2 o o o 6A3 -k Multiplying by A 4 6(6A2 28A2 124 95 95 314 814 + 31) — 8A2 + 4M +181 123 96 95 162 123 124
  2. 1 2.Show that the matrix 2 satisfies its own characteristic equation. (May 2001) 1 Solution: 1 Let A 2 equation. 2 2 The characteristic equation of given matrix is 12 — 21 + 5 = 0 1 By Cayley - Hamilton theorem, every square matrix satisfies its own characteristic 2A +51 0 (To prove) 3 4 4 3 3 2A +51 4 Hence proved 4 3 2 4 4 5 2 0 0 5 0 0 3. Using Cayley Hamilton theorem find A 4 when A 0 0 2 1 1 3 1 2 1 1 2 1 2 1 2 2 1 2 .(May Solution: For the given matrix DI —6 , D2 —8, D The characteristic equation is given by 1 3 612 + 81 By Cayley Hamilton theorem, A 3 — 6142 + 8A 31 To find A4, Multiply by A, A 4 6A3 + 2 3A 6A3 -k 8A2 29 6 22 22 124 95 95 28 23 22 23 96 95 38 28 29 162 123 124 8 7 5 5 6 6 5 9 6 7 +3 2 1 1
  3. 4. 1 Verify that A 2 (May 2003) Solution: 2 4 satisfies its own characteristic equation and hence find A 1 For the given matrix A, D —O The characteristic equation is 1 5 5 By Cayley Hamilton theorem, A 2 From this matrix A 51 Multiplying by A 4 5A2 25 o 0 25

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